On a hot grill these pizzas cook up quick. Grilling them one at a time ensures just the right amount of char on each.
Heat grill to medium-high. Toss peppers and onion with 1 tablespoon oil; grill, flipping once, 7 to 10 minutes. Combine goat cheese, Parmesan, lemon zest, and 2 teaspoons oil.
Generously brush a rimmed baking sheet with oil. Stretch one piece of dough into an 8-inch oval on sheet. Brush top with oil; season with salt. Brush grates with oil. Grill pizza, top-side down, until bubbles form, 1 to 2 minutes. Flip; dollop with a quarter of cheese mixture, peppers, and onion. Grill, moving occasionally with tongs, until cheese melts and bottom is crisp, 1 to 2 minutes. Repeat with remaining dough. Toss arugula with lemon juice and remaining 1 teaspoon oil; season with salt and pepper. Top pizzas with arugula and sprinkle with red-pepper flakes; serve.